When you first consider mathematical knowledge, and the ‘ways of knowing’ in this subject – I think the stereotype is that you rely mainly on logic and reasoning.
Considering the mathematical model, this may ring true. Beginning with axioms, you utilize deductive reasoning in order to derive new theorems which may, in turn, be used as these starting principles or axioms (creating a hierarchy of dependence of sorts with the starting axioms being the root upon which your knowledge is based upon). However, inherent within these are other ways of knowing that are perhaps not intuitively related to our base stereotypes.
For instance, a certain level of belief is required within the initial premise, the axioms that are being utilized to reach a mathematical theorem. While, due to logic, mathematical truths reached can be said to be certain (consider for instance the difference in certainty between mathematical and scientific proofs, highlighted by the difference between the term used for them – theorems vs theory’s, with one being open to negation while the other is always true given the axioms), the initial starting points require a certain level of belief. For instance, to reach the proof that a triangle adds up to 180 degrees, you may need to first believe that a straight line is 180 degrees. The other is the role/importance of imagination within Mathematics. Consider for instance Andrew Wiles and the moment when he came up with the initial idea that would eventually produce a proof for Fermat’s last theorem. The argument here is the actual origin of the idea – and whether that is the product of Wiles years of training as a mathematician, and simply a logical step that he took after exhausting multiple possibilities, or intuition (a wild idea borne in his imagination) that afforded him a chance to take the necessary deductive step in reasoning.
Something else to consider when looking at mathematical knowledge is the role of elegance or beauty to mathematics and the connections that exist to knowledge in the arts (which some argue may not even exist at all). To answer this question, the first thing that I looked into was the validity (or at least, my thoughts on the validity) of the platonic view of maths – that it is something that is discovered as opposed to invented. Certainly, the idea of certain mathematical ratios being prominent in nature – such as the golden ratio in a seashell – is something to comment on. Despite the level of abstraction that we achieve with mathematical concepts
